4
1 Some Linear Algebra
Definition 1.2.3 Let V and W be two vector spaces over the same field K . Consider
that B V = {v 1 , v 2 , . . . , v n } is a basis of V and B W = {w 1 , w 2 , . . . , w m } is a basis of
W . If T : V −→ W is a linear transformation from V into W , then the matrix of T
relative to B V and B W is the m × n matrix M = [a i j ] such that T (v j ) =
m
i=1
a i j w i .
The matrix M is written as M = [T ]
B W
B V
and it is said to be a representation of the
operator T .
The following theorem states that the study of linear transformations is, in essence,
the same as the study of matrices.
Theorem 1.2.2 Let V and W be (finite-dimensional) vector spaces over a field K
with bases B V = {v 1 , v 2 , . . . , v n } and B W = {w 1 , w 2 , . . . , w m }, respectively. Let us
denote HOM K (V, W ) the set of all linear transformations from V into W and let
Mat m×n (K ) be the set of all m × n matrices with entries in K . Then there exists a
bijection F : HOM K (V, W ) −→ Mat m×n (K ) given by F(T ) = [T ]
B W
B V
.
Exercise 1.2.3 Prove Theorem 1.2.2.
Definition 1.2.4 Given a linear transformation T : V −→ W , we define the kernel
of T , written ker(T ), as ker(T ) = {v ∈ V |T (v) = 0}. The image of T , im(T ), is the
set im(T ) = {w ∈ W |w = T (v), for some v ∈ V }.
The kernel ker(T ) is a subspace of V and im(T ) is a subspace of W . Moreover,
T is injective if and only if ker(T ) = {0}.
Theorem 1.2.3 If V is a vector space over K with dim K (V ) = dim(V ) = n < ∞,
and if T : V −→ V is a linear transformation, then the following statements are
equivalent:
(i) T is an isomorphism;
(ii) T is surjective;
(iii) T is injective.
1.3 Diagonalizable Operators
Let V be a vector space over a field K . Recall that a linear transformation T from V
into V is called a linear operator on V .
Definition 1.3.1 Let T : V −→ V be a linear operator on V . If there exists an v = 0,
v ∈ V , and λ ∈ K such that T (v) = λv, then we say that λ is an eigenvalue of T
and v is an eigenvector of T associated with λ.
The subspace V λ = {v ∈ V |T (v) = λv} of V is called eigenspace associated
with λ.
1 Some Linear Algebra
Definition 1.2.3 Let V and W be two vector spaces over the same field K . Consider
that B V = {v 1 , v 2 , . . . , v n } is a basis of V and B W = {w 1 , w 2 , . . . , w m } is a basis of
W . If T : V −→ W is a linear transformation from V into W , then the matrix of T
relative to B V and B W is the m × n matrix M = [a i j ] such that T (v j ) =
m
i=1
a i j w i .
The matrix M is written as M = [T ]
B W
B V
and it is said to be a representation of the
operator T .
The following theorem states that the study of linear transformations is, in essence,
the same as the study of matrices.
Theorem 1.2.2 Let V and W be (finite-dimensional) vector spaces over a field K
with bases B V = {v 1 , v 2 , . . . , v n } and B W = {w 1 , w 2 , . . . , w m }, respectively. Let us
denote HOM K (V, W ) the set of all linear transformations from V into W and let
Mat m×n (K ) be the set of all m × n matrices with entries in K . Then there exists a
bijection F : HOM K (V, W ) −→ Mat m×n (K ) given by F(T ) = [T ]
B W
B V
.
Exercise 1.2.3 Prove Theorem 1.2.2.
Definition 1.2.4 Given a linear transformation T : V −→ W , we define the kernel
of T , written ker(T ), as ker(T ) = {v ∈ V |T (v) = 0}. The image of T , im(T ), is the
set im(T ) = {w ∈ W |w = T (v), for some v ∈ V }.
The kernel ker(T ) is a subspace of V and im(T ) is a subspace of W . Moreover,
T is injective if and only if ker(T ) = {0}.
Theorem 1.2.3 If V is a vector space over K with dim K (V ) = dim(V ) = n < ∞,
and if T : V −→ V is a linear transformation, then the following statements are
equivalent:
(i) T is an isomorphism;
(ii) T is surjective;
(iii) T is injective.
1.3 Diagonalizable Operators
Let V be a vector space over a field K . Recall that a linear transformation T from V
into V is called a linear operator on V .
Definition 1.3.1 Let T : V −→ V be a linear operator on V . If there exists an v = 0,
v ∈ V , and λ ∈ K such that T (v) = λv, then we say that λ is an eigenvalue of T
and v is an eigenvector of T associated with λ.
The subspace V λ = {v ∈ V |T (v) = λv} of V is called eigenspace associated
with λ.